Name: signal symmetry and decomposition
Uploaded: 2024-09-26T13:16:53.000Z
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K12 Standards

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South Carolina College and Career Ready Science Standards

High School

Physics

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Waves and Their Applications in Technologies for Information Transfer

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Even and Odd Signals

Un signal pair, en temps continu ou en temps discret, est défini comme un signal qui correspond à sa propre version inversée dans le temps.
Les signaux pairs sont symétriques autour de l’axe vertical, ce qui signifie que le signal pour les valeurs temporelles négatives reflète le signal pour les valeurs temporelles positives.
Un signal est dit impair s’il équivaut au négatif de son homologue inversé dans le temps.
Les signaux étranges sont reconnus par leur nature antisymétrique autour de l’axe vertical.
Tout signal en temps continu peut être exprimé sous la forme d’une combinaison de composantes paires et impaires.
Ici, le premier composant du côté droit est une fonction paire, tandis que le deuxième composant est impair.
Un signal complexe peut être décomposé en composantes paires et impaires à l’aide de symétries conjuguées.
Le produit d’une fonction paire et d’une fonction impaire est une fonction impaire. La multiplication de deux fonctions du même type, paires ou impaires, donnera une fonction paire.
Enfin, l’addition ou la soustraction de deux fonctions paires produira une fonction paire, et de même, l’addition ou la soustraction de deux fonctions impaires donnera une fonction impaire.

signaux pairs et impairs

Un signal pair, qu'il soit à temps continu ou discret, est défini par sa symétrie avec sa version inversée dans le temps. Mathématiquement, cela se traduit par


<img alt="Equation1" src="/files/ftp_upload/16039/16039_PT%20EQUATION_1.svg" style="height: 14px;">

pour les signaux à temps continu et


<img alt="Equation2" src="/files/ftp_upload/16039/16039_PT%20EQUATION_2.svg" style="height: 14px;">

pour les signaux à temps discret. Les signaux pairs présentent une symétrie autour de l'axe vertical, ce qui signifie que le signal pour les valeurs de temps négatives reflète celui pour les valeurs de temps positives.


En revanche, un signal est dit impair s'il ne correspond pas à son homologue inversé dans le temps, représenté par


<img alt="Equation3" src="/files/ftp_upload/16039/16039_PT%20EQUATION_3.svg" style="height: 14px;">

pour les signaux à temps continu et


<img alt="Equation4" src="/files/ftp_upload/16039/16039_PT%20EQUATION_4.svg" style="height: 14px;">

pour les signaux à temps discret. Les signaux impairs sont caractérisés par leur nature antisymétrique autour de l'axe vertical.


Tout signal continu peut être exprimé comme une combinaison de composantes paires et impaires. Cette décomposition est donnée par :


<img alt="Equation5" src="/files/ftp_upload/16039/16039_PT%20EQUATION_5.svg" style="height: 14px;">

Ici, le premier terme du côté droit est une fonction paire, tandis que le second terme est une fonction impaire.


Les signaux complexes peuvent également être décomposés en composantes paires et impaires à l'aide des symétries conjuguées. Le produit d'une fonction paire et d'une fonction impaire donne une fonction impaire. De plus, la multiplication de deux fonctions du même type, soit toutes deux paires, soit toutes deux impaires, donne une fonction paire.


De plus, les opérations algébriques d'addition et de soustraction suivent des règles spécifiques : l'addition ou la soustraction de deux fonctions paires donne une fonction paire, et l'addition ou la soustraction de deux fonctions impaires produit une fonction impaire. Ces propriétés sont fondamentales dans l'analyse des signaux, car elles permettent la décomposition et la simplification de signaux complexes en éléments facilement manipulables.

Signaux pairs et impairs

Un signal pair, en temps continu ou en temps discret, est défini comme un signal qui correspond à sa propre version inversée dans le temps.
Même les signaux sont ...

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Electrical Engineering

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Unit 1

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Introduction to Signals and Systems

Introduction aux signaux et systèmes

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

signal symmetry and decomposition

Signal symmetry helps classify continuous-time and discrete-time signals as even or odd. An even signal matches its time-reversed version. It mirrors around the vertical axis, so the values at negative time match the values at positive time.

An odd signal does not match its time-reversed counterpart. It has antisymmetry about the vertical axis. These symmetry rules are used in both continuous-time and discrete-time signal analysis.

Any continuous-time signal can be split into even and odd parts. The decomposition writes the signal as a sum of one even function and one odd function. Complex signals can also be separated into even and odd components by using conjugate symmetry.

These signal types also follow clear algebra rules. The product of an even function and an odd function is odd. The product of two functions of the same type, whether both even or both odd, is even. Addition and subtraction also stay consistent: even with even gives even, and odd with odd gives odd. These patterns make complex signals easier to analyze and simplify.

Signal Symmetry and Decomposition

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Signals and Systems in Everyday Data

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even and odd signals

Signal symmetry helps classify continuous-time and discrete-time signals as even or odd. An even signal matches its time-reversed version. It mirrors around the vertical axis, so the values at negative time match the values at positive time. An odd signal does not match its time-reversed counterpart. It has antisymmetry about the vertical axis. These symmetry rules are used in both continuous-time and discrete-time signal analysis. Any continuous-time signal can be split into even and odd...

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Step, Impulse, and Ramp Signals

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rectangular and triangular pulse function

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Modeling Pulses with Rectangles and Triangles

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exponential and sinusoidal signals

Complex Exponentials in Signal Behavior

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basic discrete time signals

Discrete-Time Signal Basics and Shapes

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Signal Transformations in Time and Amplitude

Signal transformations in time and amplitude are key tools in signal processing. They include time reversal, time scaling, time shifting, and amplitude changes. These operations help students describe how a continuous-time signal changes when it is mirrored, stretched, moved, or resized. Time reversal flips a continuous-time signal about the vertical axis at t = 0. It is done by replacing t with -t, so x(t) becomes x(-t). A graph of the signal shows this mirrored shape clearly. Time scaling...

classification of systems-i

System Properties: Linearity, Causality, Memory

System properties such as linearity, causality, and memory help describe how a system responds to input. These ideas are useful for studying and designing systems in engineering. A system can be checked by looking at how its output changes with different inputs. Linearity means the input-output relationship follows homogeneity and additivity. Homogeneity says that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Additivity means the response...

classification of systems-ii

Continuous and Discrete System Models

Continuous-time and discrete-time systems are two common ways to model signals and their outputs. In continuous-time systems, both input and output signals change smoothly as time moves continuously. These systems are often written with differential or algebraic equations. An RC circuit is one example of a continuous-time system. Its input and output voltage can be related by a differential equation based on Ohm’s law and the capacitor relation. Discrete-time systems work differently. Their...

key terms and concepts

Learn how even and odd signals are identified by symmetry, how any signal can be split into even and odd parts, and the rules for combining them.

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Signal Symmetry and Decomposition

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